Periodic orbits in the Kepler-Heisenberg problem

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June 2014, 6(2): 261-278. doi: 10.3934/jgm.2014.6.261

Periodic orbits in the Kepler-Heisenberg problem

Corey Shanbrom ^1,

1.	Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA, United States

Received November 2013 Revised March 2014 Published June 2014

Abstract
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One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.

Keywords: Heisenberg group, Periodic orbits, fundamental solution to Laplacian., integrable system, Kepler problem, Carnot group.

Mathematics Subject Classification: 53C17, 37N05, 37J45, 70H12, 70H0.

Citation: Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261

References:

[1]	R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978.
[2]	A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2. doi: 10.1134/S156035470806004X.
[3]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.
[4]	D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416.
[5]	G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744. doi: 10.2307/2370714.
[6]	O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960.
[7]	R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27.
[8]	A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.
[9]	F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266. doi: 10.1007/s00332-011-9116-z.
[10]	G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4.
[11]	I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963.
[12]	W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.
[13]	P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983.
[14]	N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow.
[15]	R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002.
[16]	R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere, to appear in Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, Fields Institute Communications series.
[17]	R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30. doi: 10.1007/BF01941322.
[18]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.
[19]	C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013.
[20]	L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980.

show all references

References:

[1]	R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978.
[2]	A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2. doi: 10.1134/S156035470806004X.
[3]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.
[4]	D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416.
[5]	G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744. doi: 10.2307/2370714.
[6]	O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960.
[7]	R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27.
[8]	A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.
[9]	F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266. doi: 10.1007/s00332-011-9116-z.
[10]	G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4.
[11]	I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963.
[12]	W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.
[13]	P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983.
[14]	N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow.
[15]	R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002.
[16]	R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere, to appear in Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, Fields Institute Communications series.
[17]	R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30. doi: 10.1007/BF01941322.
[18]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.
[19]	C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013.
[20]	L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980.

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